A Minimum Problem for Finite Sets of Real Numbers with Nonnegative Sum

G. Chiaselotti, G. Marino and C. Nardi

Journal of Applied Mathematics, 2012, vol. 2012, 1-15

Abstract:

Let 𠑛 and 𠑟 be two integers such that 0 < 𠑟 ≤ 𠑛 ; we denote by 𠛾 ( 𠑛 , 𠑟 ) [ 𠜂 ( 𠑛 , 𠑟 ) ] the minimum [maximum] number of the nonnegative partial sums of a sum ∑ 𠑛 1 = 1 𠑎 𠑖 ≥ 0 , where 𠑎 1 , … , 𠑎 𠑛 are 𠑛 real numbers arbitrarily chosen in such a way that 𠑟 of them are nonnegative and the remaining 𠑛 − 𠑟 are negative. We study the following two problems: ( 𠑃 1 ) which are the values of 𠛾 ( 𠑛 , 𠑟 ) and 𠜂 ( 𠑛 , 𠑟 ) for each 𠑛 and 𠑟 , 0 < 𠑟 ≤ 𠑛 ? ( 𠑃 2 ) if 𠑞 is an integer such that 𠛾 ( 𠑛 , 𠑟 ) ≤ 𠑞 ≤ 𠜂 ( 𠑛 , 𠑟 ) , can we find 𠑛 real numbers 𠑎 1 , … , 𠑎 𠑛 , such that 𠑟 of them are nonnegative and the remaining 𠑛 − 𠑟 are negative with ∑ 𠑛 1 = 1 𠑎 𠑖 ≥ 0 , such that the number of the nonnegative sums formed from these numbers is exactly 𠑞 ?

Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnljam:847958

DOI: 10.1155/2012/847958

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